Robust safety-critical control of nonlinear systems with small perturbations

This paper studies the robust safety-critical control problem for a class of nonlinear systems with small perturbations on a prescribed safe set. First, a formal definition of robust safety is presented, which means safety of nonlinear systems with small perturbations on the originally prescribed safe set. Then, robust safety-critical control problem is investigated, with the aim to ensure that nonlinear systems with small perturbations are robust safety and state trajectories for perturbed systems starting from the prescribed safe set can keep arbitrarily close to the equilibrium for unperturbed systems. To solve robust safety-critical control problem of perturbed systems on the originally prescribed safe set, the robust safety problem and the robust stability problem are analyzed on the originally safe set, respectively. Meanwhile, a quadratic program is used to obtain a synthetic robust safety-critical controller by combining control barrier functions and control Lyapunov functions of unperturbed systems, which can simultaneously guarantee robust safety and robust stability for perturbed systems on the originally prescribed safe set. In addition, the corresponding results can be extended to cascade-connected nonlinear systems. Finally, three illustrative examples are provided to demonstrate effectiveness of the proposed results.